Properties of linear transformation

Properties of linear transformation

Lessons

Recall from last chapter the 2 properties of AxAxAx:

1. A(u+v)=Au+AvA(u+v)=Au+AvA(u+v)=Au+Av

2. A(cu)=c(Au)A(cu)=c(Au)A(cu)=c(Au)

where uuu and vvv are vectors in Rn\Bbb{R}^nR​n​​ and ccc is a scalar.

Now the properties of linear transformation are very similar. Linear transformation preserves the operations of vector addition/subtraction and scalar multiplication. In other words, If T is linear, then:

1. T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)T(u+v)=T(u)+T(v)

2. T(cu)=cT(u)T(cu)=cT(u)T(cu)=cT(u)

3. T(0⃗)=0⃗T(\vec{0})=\vec{0} T(​0​⃗​​)=​0​⃗​​

We can even combine property 1 and 2 to show that:

T(cu+dv)=cT(u)+dT(v)T(cu+dv)=cT(u)+dT(v) T(cu+dv)=cT(u)+dT(v)

where uuu, vvv are vectors and ccc, ddd are scalars. Note that if this equation holds, then it must be linear.

If you have more than 2 vectors and 2 scalars? What if you have p vectors and p scalars? Then we can generalize this equation and say that:

How to see if a transformation is linear
• Show that: T(cu+dv)=cT(u)+dT(v)T(cu+dv)=cT(u)+dT(v)T(cu+dv)=cT(u)+dT(v)
• General formula: T(c1v1+c2v2+⋯+cnvn)=c1T(v1)+c2T(v2)+⋯+cpT(vp)T(c_1 v_1+c_2 v_2+\cdots+c_n v_n )=c_1 T(v_1 )+c_2 T(v_2 )+\cdots+c_p T(v_p)T(c​1​​v​1​​+c​2​​v​2​​+⋯+c​n​​v​n​​)=c​1​​T(v​1​​)+c​2​​T(v​2​​)+⋯+c​p​​T(v​p​​)

1.

Understanding and Using the Properties
Show that the transformation TTT defined by is not linear.

2.

Show that the transformation TTT defined by is not linear.

3.

Proving Questions using the Properties
An affine transformation T:RnT: \Bbb{R}^n T:R​n​​→Rm \Bbb{R}^mR​m​​ has the form T(x)=Ax+bT(x)=Ax+bT(x)=Ax+b, where AAA is an m×nm \times nm×n matrix and bbb is a vector in Rn\Bbb{R}^nR​n​​. Show that the transformation TTT is not a linear transformation when b≠0b \neq 0b≠0.

4.

Define T:RnT: \Bbb{R}^n T:R​n​​→Rm \Bbb{R}^mR​m​​ to be a linear transformation, and let the set of vectors {v1,v2,v3v_1,v_2,v_3v​1​​,v​2​​,v​3​​ } be linearly dependent. Show that the set of vectors {T(v1),T(v2),T(v3)T(v_1),T(v_2),T(v_3)T(v​1​​),T(v​2​​),T(v​3​​)} are also linearly dependent.

5.

Define T:RnT: \Bbb{R}^n T:R​n​​→Rm \Bbb{R}^mR​m​​ to be a linear transformation and the set of vectors v1v_1v​1​​,...,vpv_pv​p​​ are in Rn\Bbb{R}^nR​n​​. In addition, let T(vi)=0T(v_i )=0T(v​i​​)=0 for i=1,2,i=1,2,i=1,2,…,p,p,p. If xxx is any vector in Rn\Bbb{R}^nR​n​​, then show that T(x)=0T(x)=0T(x)=0. In other words, show that TTT is the zero transformation.